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Twenty years ago, Woodward and Byrd (1983) argued that,
despite the presence of area and perimeter in the U.S. math-
ematics curriculum, something was quite wrong with
students’understanding of the topics. They gave eighth-
grade students (thirteen to fourteen years old) a task:
Mr. Young had 60 feet of fencing available to enclose a
garden. He wanted the garden to be rectangular in shape.
Also, he wanted to have the largest possible garden area.
He drew a picture of several possibilities for the gar-
den, each with a perimeter of 60 feet (see Figure 1).
Consider Mr. Young’s drawings of the garden plots.
Check the statement below that he found to be true.
Garden I is the biggest garden
Garden II is the biggest garden
Garden III is the biggest garden
Garden IV is the biggest garden
Garden V is the biggest garden
These gardens are all the same size. (p. 344)
Nearly 60% of these students chose, “These gardens are all
the same size.” The researchers concluded that the students
had confused area with perimeter. More fundamentally, they
argued that area and perimeter at that time were topics cov-
ered in classrooms (as they certainly are today!), but not
concepts taught. That is, students were given definitions and
formulas, but generally emerged with no understanding of
what area and perimeter really meant, nor how they related
to each other. They argued that the students in their study,
in being unable to distinguish area from perimeter, were typ-
ical U.S. students and that this was likely to be the fault of
the curriculum and its instruction.
The last fifty years have seen roughly three paradigms
for school mathematics (K-12 – kindergarten to twelfth
grade or students aged 5 to 18) in the U.S.:
•the New Math of the 1950s and 1960s, which
emphasized the structures of the discipline of
mathematics such as axioms, proofs and precise
definitions
•the traditional curriculum, in which algorithms and
skills took precedence
•the reform curriculum, where the stated goal is to
increase the number of students who see the sub-
ject as relevant, useful and beautiful.
The topics in the curriculum vary some across the para-
digms, but their treatments vary widely. This article analyzes
three curricular approaches to one topic – perimeter – with
an emphasis on the approach of one reform curriculum.
Test results seem to support the contention that concep-
tual learning of area and perimeter rarely happens in U.S.
classrooms (e.g., Carpenter et al., 1981). Students often
choose answers that correspond to superficial aspects of a
task – adding all the numbers in a diagram, for instance,
without regard to the prompt. Yet this performance is in no
way attributable to an absence of area and perimeter from the
middle school (ages12 to 14) curriculum. On the contrary,
formulas for area and perimeter of rectangles are ubiquitous
in textbooks. The current curricular reform seeks to address
the lack of depth in student understanding of these topics by
focusing on the concepts rather than on the formulas – teach-
ing for understanding. This leaves open the question of what
a deep understanding of perimeter might look like.
What is perimeter?
The standard conception of perimeter is, as the Greek root
would suggest, distance around. But distance around may
not tell the whole story of perimeter. Look at the Punctured
square [1] (see Figure 2). What is its perimeter? There are
arguably three reasonable answers [2] to the question of this
figure’s perimeter. I will present each as the logical conclu-
sion based on a development of school curriculum.
Answer 1: This figure has no perimeter
According to the archetypal New Math curriculum, written
by the School Mathematics Study Group (SMSG, 1965),
perimeter is not defined for the Punctured square. SMSG
develops perimeter in second grade (age 7) as part of the
study of linear measure (SMSG, 1965). Students first mea-
sure the lengths of line segments, both directly and by
measuring a string with the same length. They then move
on to using a string to measure the length of a curve. Finally,
the curve is closed and perimeter is defined as the “length
of a polygon” (ibid. p. 526). The Punctured square is not a
polygon. Therefore, perimeter is not defined for it.
Mathematically, the distinction here is fundamental. The
two remaining curricular approaches treat this one figure as
PERIMETER IN THE CURRICULUM
CHRISTOPHER DANIELSON
For the Learning of Mathematics 25, 1 (March, 2005)
FLM Publishing Association, Edmonton, Alberta, Canada
Figure 1: Several possibilities for the garden.
